\documentclass{article} %\usepackage{amsmath,amsthm,amssymb} \usepackage{amsmath,amssymb} \begin{document} \pagestyle{empty} {\large Complexity in low-dimensional topology} Let $\cal O$ be a class of geometric objects. By a {\em complexity function} on $\cal O$ we mean a map $c\colon O\to {\Bbb N}\cup \{ 0\}$. Of course, we expect that $c$ is natural, i.e. it measures how complicated a combinatorial description of a given object must be. Using $c$, we can potentially do the following: \begin{enumerate} \item Convert observations into rigorous statements. \item Classify objects of small complexity. \item Use inductive arguments. \end{enumerate} There are several ways to construct reasonable complexity functions. First, one can try to count the number of singularities of a given object $A\in \cal O$. For example, a natural complexity of a graph will probably contain information on the number of its vertices. But what to do if the objects in question are homogenous, like manifolds? Then one can count singularities of different presentations of a given object and take the minimum. For example, knots are usually tabulated according to the number of double points of their minimal projections. Based on this idea, we introduce different complexity functions for 3-manifolds, compare them, and describe their properties. One of them ({\em spine complexity}) is very close to the number of tetrahedra in a minimal triangulation, the other ({\em Heegaard complexity}) takes into account the number of crossing points of the meridians that form a minimal Heegaard diagram. As computer experiments show, the number of 3-manifolds having complexity $n$ grows exponentially in the first case and only polynomially in the second one. We introduce also an {\em extended complexity}, which is a universal tool for inductive proofs. \end{document}